Normalization

For any fixed values of $ℓ$ and $s$, the spin-weighted spherical harmonics normalized as usual, satisfy the relation

\[\sum_{m} |{}_{s}Y_{ℓ,m}(𝐐)|² = \frac{2ℓ+1}{4π},\]

for every $𝐐 ∈ \mathrm{Spin}(3)$. This result is crucial for relating values of the spin-weighted spherical harmonics to Wigner $𝔇$ matrices, and we will derive it here.

Begin by fixing $ℓ$ and $s$, and define the space of functions with these eigenvalues:

\[ℋ_{ℓ,s} = \left\{ f : \mathrm{Spin}(3) → ℂ \middle| L² f = ℓ(ℓ+1) f \mathrm{\ \ and\ \ } R_z f = s f \right\}.\]

Now, for a given point $𝐏 ∈ \mathrm{Spin}(3)$, we define the "reproducing kernel" $K_𝐏$ as a function in $ℋ_{ℓ,s}$ such that for every square-integrable function $f ∈ ℋ_{ℓ,s}$,

\[f(𝐏) = \int_{\mathrm{Spin}(3)} K̄_𝐏(𝐐)\, f(𝐐)\, d𝐐.\]

Now, we want to expand this kernel in terms of the basis functions ${}_{s}Y_{ℓ,m}$ of this space. Recall that these are not orthonormal over $\mathrm{Spin}(3)$; they are only orthogonal. They are normalized over $𝕊²$ in the restricted sense discussed here so that, when integrating over $\mathrm{Spin}(3)$, we get an extra factor of $π/2$:

\[\int_{\mathrm{Spin}(3)} {}_{s}Ȳ_{ℓ,m'}(𝐐)\, {}_{s}Y_{ℓ,m}(𝐐)\, d𝐐 = \frac{π}{2} δ_{m',m}.\]

Now, if we expand $K_𝐏$ and $f$ in terms of these basis functions, we can calculate

\[\begin{aligned} f(𝐏) &= \int_{\mathrm{Spin}(3)} K̄_𝐏(𝐐)\, f(𝐐)\, d𝐐 \\ &= \int_{\mathrm{Spin}(3)} \sum_{m',m} K̄_{𝐏,m'}\, {}_{s}Ȳ_{ℓ,m'}(𝐐)\, f_{m}\, {}_{s}Y_{ℓ,m}(𝐐)\, d𝐐 \\ &= \frac{π}{2} \sum_{m} K̄_{𝐏,m}\, f_{m}, \end{aligned}\]

the last of which implies that

\[K_{𝐏,m} = \frac{2}{π} {}_{s}Ȳ_{ℓ,m}(𝐏)\]

for every $m$. That is,

\[K_𝐏(𝐐) = \sum_m \frac{2}{π} {}_{s}Ȳ_{ℓ,m}(𝐏)\; {}_{s}Y_{ℓ,m}(𝐐).\]

Now we take the norm of the kernel function:

\[\begin{aligned} \|K_𝐏\|²_{\mathrm{Spin}(3)} &= \int_{\mathrm{Spin}(3)} |K_𝐏(𝐐)|²\, d𝐐 \\ &= \int_{\mathrm{Spin}(3)} \sum_{m',m} \frac{4}{π²} {}_{s}Y_{ℓ,m'}(𝐏)\, {}_{s}Ȳ_{ℓ,m'}(𝐐)\, {}_{s}Ȳ_{ℓ,m}(𝐏)\, {}_{s}Y_{ℓ,m}(𝐐)\, d𝐐 \\ &= \frac{2}{π} \sum_{m} |{}_{s}Y_{ℓ,m}(𝐏)|². \end{aligned}\]

Now, we are integrating with respect to a Haar measure on the group $\mathrm{Spin}(3)$, which means that the integral must be invariant under group actions. In particular, this means that the norm of the kernel function cannot depend on the choice of $𝐏$, which means that the sum in the last expression is independent of $𝐏$:

\[\sum_{m} |{}_{s}Y_{ℓ,m}(𝐏)|² = \sum_{m} |{}_{s}Y_{ℓ,m}(𝐏')|²\]

for all $𝐏, 𝐏' ∈ \mathrm{Spin}(3)$.

The last trick is to just integrate this expression over $\mathrm{Spin}(3)$ again, and evaluate it in two different ways. On one hand, we have

\[\int_{\mathrm{Spin}(3)} \sum_{m} |{}_{s}Y_{ℓ,m}(𝐐)|²\, d𝐐 = \sum_{m} \int_{\mathrm{Spin}(3)} |{}_{s}Y_{ℓ,m}(𝐐)|²\, d𝐐 = \sum_{m} \frac{π}{2} = (2ℓ+1) \frac{π}{2},\]

since there are $2ℓ+1$ values of $m$ in the sum. On the other hand, since the sum is independent of $𝐐$, we can change to integrating over a dummy variable and pull the sum out of the integral:

\[\int_{\mathrm{Spin}(3)} \sum_{m} |{}_{s}Y_{ℓ,m}(𝐐)|²\, d𝐐 = \sum_{m} |{}_{s}Y_{ℓ,m}(1)|²\, \int_{\mathrm{Spin}(3)} d𝐐' = 2π² \sum_{m} |{}_{s}Y_{ℓ,m}(1)|².\]

Equating these two expressions, we find that

\[\sum_{m} |{}_{s}Y_{ℓ,m}(𝐐)|² = \frac{2ℓ+1}{4π},\]

for arbitrary $ℓ$, $s$, and $𝐐 ∈ \mathrm{Spin}(3)$.

Now, we use this to relate the spin-weighted spherical harmonics to the Wigner $𝔇$ matrices. Recall that the SWSHs are defined in terms of their eigenvalues: ${}_{s}Y_{ℓ,m}$ has $L_z$ eigenvalue $m$ and $R_z$ eigenvalue $s$. The $𝔇$ matrices have eigenvalues $L_z 𝔇^{(ℓ)}_{m',m} = -m'\, 𝔇^{(ℓ)}_{m',m}$ and $R_z 𝔇^{(ℓ)}_{m',m} = m\, 𝔇^{(ℓ)}_{m',m}$, so it is the conjugate $\overline{𝔇^{(ℓ)}_{m, -s}}$ that has $L_z$ eigenvalue $m$ and $R_z$ eigenvalue $s$. Both functions therefore lie in the same one-dimensional eigenspace, so

\[{}_{s}Y_{ℓ,m}(𝐐) = κ_{ℓ,s}\, \overline{𝔇^{(ℓ)}_{m, -s}(𝐐)}\]

for some constant $κ_{ℓ,s}$ independent of $𝐐$ and of $m$ — the $L_±$ ladder operators are linear, so they commute with the map between $Y$ and $\bar{𝔇}$ and force the same constant for every $m$.

To find its magnitude, use the unitarity of $𝔇^{(ℓ)}$, which makes each of its columns a unit vector:

\[\sum_{m} |{}_{s}Y_{ℓ,m}(𝐐)|² = \frac{2ℓ+1}{4π} = |κ_{ℓ,s}|² \sum_{m} |𝔇^{(ℓ)}_{m, -s}(𝐐)|² = |κ_{ℓ,s}|².\]

Thus, up to a phase factor,

\[|κ_{ℓ,s}| = \sqrt{\frac{2ℓ+1}{4π}}.\]

At $s = 0$ that phase factor is 1, because of the near-universal agreement that $Y_{ℓ,0}(θ, ϕ)$ should be real-valued, with the sign chosen so that it is positive at the north pole $θ = 0$. We interpret that to mean that ${}_{0}Y_{ℓ,0}(𝟏)$ is real and positive, and since ${}_{0}Y_{ℓ,0}(𝟏) = κ_{ℓ,0}\, \overline{𝔇^{(ℓ)}_{0,0}(𝟏)}$ with $𝔇^{(ℓ)}_{m', m}(𝟏) = δ_{m',m}$, the constant must be real and positive.

The dependence on $s$ is then fixed by the $R_±$ ladder operators. Conjugation reverses them — a short calculation from the definitions gives $R_+ \bar{g} = -\overline{R_- g}$ — so

\[R_+ \left\{\overline{𝔇^{(ℓ)}_{m, -s}}\right\} = -\overline{R_- 𝔇^{(ℓ)}_{m, -s}} = -\sqrt{(ℓ-s)(ℓ+s+1)}\; \overline{𝔇^{(ℓ)}_{m, -(s+1)}},\]

while the convention for the SWSHs is $R_+ \left\{{}_{s}Y_{ℓ,m}\right\} = \sqrt{(ℓ-s)(ℓ+s+1)}\, {}_{s+1}Y_{ℓ,m}$ with a positive coefficient. Comparing the two gives $κ_{ℓ,s+1} = -κ_{ℓ,s}$, and so, with the value at $s = 0$ already fixed, this completely determines the relationship between the spin-weighted spherical harmonics and the Wigner $𝔇$ matrices:

\[{}_{s}Y_{ℓ,m}(𝐐) = (-1)^s \sqrt{\frac{2ℓ+1}{4π}}\; \overline{𝔇^{(ℓ)}_{m, -s}(𝐐)}\]

for every $ℓ$, $m$, $s$, and $𝐐$. This is the definition recorded in the conventions summary. (For half-integer $s$ the factor $(-1)^s$ means the principal branch $e^{iπs}$.)